The first disclaimer is that the pup I'm using for reference is a Gen 2 with differing color codes and (what I assume is) shielding paint on the interior side of the baseplate. The second disclaimer is that, if the presumed short is occurring due to a break in the windings in either of the bobbins, you're typically faced with having to rewind it. Assuming the least, this post will address some other potential causes:
Once the two mounting polepieces and four phillips screws which secure the bobbin to the baseplate are out of the way, you'll see the soldering pad to which the gray grounding wire should be attached (upper left of photo 1). Note that wax adhesion may require you to gently pry the baseplate from the rest of the assembly.
In Gen 1 pickups (or at least the ones I've disassembled), the PCB-style baseplate is entirely nonferromagnetic; meaning the grounding wire is largely for the sake of preventing the polepieces from buzzing when touched. I mention that because the loose grounding wire in chrisjscott's pickup would only be causing the described issues
if its frayed strands were causing a short by coming into contact with one of the pickup's other bare wires. So if you use a razor to carefully cut the section of shrink-tubing bundling the five wires where they converge at the pickup, you should be able to get some extra slack from the length of the grounding wire to resolder it to its pad on the baseplate.
In the next photo, we have the point where each coil's start and finish wires are soldered to the color-coded ones. Being as the pickup I'm using for reference is working as it should, I didn't cut away the insulating pickup tape covering the windings to show exactly where they're soldered together. But checking those connections for any breaks or fraying strands shorting themselves out will help diagnose whether the problem lies within this junction or within the windings themselves.
The third photo is me attempting to show how the pickup leads are fitted to lie in a channel formed between the ferromagnetic pickup spacer and the edge of the bobbins' winding tape. The baseplate design actually leaves you plenty of room to reassemble the pickup without the wires being perfectly back in place.
The fourth photo is intended to show the best place for cutting the pickup's insulating tape; where the space between the two bobbins means the tape isn't in contact with any windings. It also happens to be where the windings' junction to the color-coded wires are in Dimarzio Fly pickups; which means easier access for troubleshooting.
Lastly, we have our reassembled pickup. I didn't have enough hands to photograph myself using the multimeter - But suffice to say it confirmed the pup's clean bill of health. If you ended up cutting the pickup's insulating tape, a few wraps of common black electrical tape around the pickup should do the trick.
Very helpful! 
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